An object of mass $8 \; kg$ is whirled round in a vertical circle of radius $2 \; m$ with a constant speed of $6 \; m/s$. Calculate the maximum and the minimum tension in the string.

Given,

Mass of the object $(m) = 8\;kg$

Radius of the object $(r) = 2\;m$

Speed of the object $(v) = 6\;m/s$

Maximum tension $(T_{max}) = \;?$

Minimum tension $(T_{min}) = \;?$

We have,

$T_{max} = \frac{m\;v^2}{r} + m\;g = \frac{8 \; * \; 6^2}{2} + 8\;*\;10 = 224\;N$

∴ Maximum tension $(T_{max}) = \;224\;N$

$T_{min} = \frac{m\;v^2}{r} - m\;g = \frac{8 \; * \; 6^2}{2} - 8\;*\;10 = 64\;N$

∴ Minimum tension $(T_{min}) = \;64\;N$

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A mass of $0.2 \; kg$ is rotated by a string at a constant speed in a vertical circle of radius $1 \; m$. If the minimum tension int he string is $3 \; N$. Calculate the magnitude of the speed and the maximum tension in the string.

Given,

Mass of the stone $(m) = 0.2\;kg$

Radius of the circle $(r) = 1\;m$

Minimum Tension in String $(T_{min}) = 3\;N$

Magnitude of the speed $(v) = \;?$

Maximum tension in the string $(T_{max}) = \;?$

We have,

$T_{max} = \frac{m\;v^2}{r} + mg$

$T_{min} = \frac{m\;v^2}{r} - mg$

From equation (ii); we get,

$v^2 = \frac{T_{min}\;*\;r}{m} + m\;g = \frac{3\;*\;1}{0.2} + 0.2\;*\;10 = 25$

⇒ $v = 5\;m/s$

∴ Magnitude of the speed $(v) = \;5\;m/s$

From equation (i), we get,

$T_{max} = \frac{0.2\;*\;5^2}{1} + 0.2\;*\;10 = 7\;N$

∴ Maximum tension in the string $(T_{max}) = \;7\;N$

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A object of mass $4 \; kg$ is whirled round a vertical circle of radius $1 \; m$ with a constant speed of $3 \; m/s$. Calculate the maximum tension in the string.

Given,

Mass of the object $(m) = 4\;kg$

Radius of the object $(r) = 1\;m$

Speed of the object $(v) = 3\;m/s$

Maximum tension $(T_{max}) = \;?$

We have,

$T_{max} = \frac{m\;v^2}{r} + m\;g = \frac{4 \; * \; 3^2}{1} + 4\;*\;10 = 76\;N$

Maximum tension $(T_{max}) = \;76\;N$

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